Problem 25 - Algebra and Trigonometry

Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=\sqrt{3} ; \quad r=\sqrt{3} $$

Step-by-Step Solution

Answer

The fifth term is \(9\sqrt{3}\). The nth term is \(a_n = \sqrt{3} \times 3^{(n-1)/2}\).

Step 1: Formula for the nth term of a geometric sequence

The nth term of a geometric sequence can be found using the formula: \[ a_n = a_1 \times r^{(n-1)} \] where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Find the fifth term (\(a_5\))

To find the fifth term, substitute \(a_1 = \sqrt{3}\), \(r = \sqrt{3}\), and \(n = 5\) into the nth term formula:\[ a_5 = \sqrt{3} \times (\sqrt{3})^{(5-1)} \]\[ a_5 = \sqrt{3} \times (\sqrt{3})^4 \]Simplify \((\sqrt{3})^4\):\[ (\sqrt{3})^4 = 3^2 = 9 \]Now multiply:\[ a_5 = \sqrt{3} \times 9 = 9\sqrt{3} \]

Step 3: General formula for the nth term

Substitute the given values \(a_1 = \sqrt{3}\) and \(r = \sqrt{3}\) into the general formula:\[ a_n = \sqrt{3} \times (\sqrt{3})^{(n-1)} \] Simplify the exponent:\[ a_n = \sqrt{3} \times 3^{(n-1)/2} \]So the nth term formula for this geometric series is:\[ a_n = \sqrt{3} \times 3^{(n-1)/2} \]

Key Concepts

nth term formula

To understand geometric sequences, we need to start with the nth term formula. This formula helps us find the term at any position in a geometric sequence. The nth term formula is given by:

\( a_n = a_1 \times r^{(n-1)} \).

Here:

• \( a_n \) is the term you are trying to find,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio, and
• \( n \) is the term number.

In our problem, the first term \( a_1 \) is \( \sqrt{3} \) and the common ratio \( r \) is also \( \sqrt{3} \). By substituting these values into the formula, you can find any term in the sequence.

common ratio

The common ratio, \( r \), is a key part of any geometric sequence. It is the factor by which you multiply each term to get the next term. If you know the common ratio and the first term, you can find all terms in the sequence.

In this exercise, \( r = \sqrt{3} \). So, if our first term \( a_1 \) is \( \sqrt{3} \), the second term will be \( \sqrt{3} \times \sqrt{3} = 3 \).

The third term will be \( 3 \times \sqrt{3} = 3\sqrt{3} \), and so on. The common ratio remains consistent throughout, and it dictates the pattern of the sequence.
Once you understand the common ratio, predicting the next terms becomes straightforward.

geometric series

A geometric series is the sum of the terms in a geometric sequence. If we were to sum the first few terms of our sequence, we would be calculating a geometric series.

For instance, with the first few terms being \( \sqrt{3}, 3, 3\sqrt{3} \), the sum would be \( \sqrt{3} + 3 + 3\sqrt{3} \).

Finding the sum of a geometric series, especially if it has many terms, involves specific formulas. The sum to the nth term, \( S_n \), can be given by: \[ S_n = a_1 \frac{1-r^n}{1-r} \]\ if \( r \eq 1 \).

Using this formula makes it easy to find the sum of multiple terms without adding one by one. Mastering this concept is essential for tackling more complex mathematical problems involving sequences.